A significant problem that product retailers face is determining the best plan for liquidating inventory while maximizing revenue during a given sales period. For example, a fashion retailer may seek to maximize profits for a given clothing line during a predetermined sales season while meeting an inventory sales target, where only a certain percentage of beginning inventory remains unsold at the end of the season. This problem is sometimes referred to as a markdown optimization problem in retail practice and can involve identifying which products to mark down, how much to mark the products down, when to mark down the products, and/or in which markets and stores the markdowns should be taken.